Question about subset

PhysicsGente

Hello, I was looking into this proof

http://www.proofwiki.org/wiki/Lipsch...lly_Equivalent

and I was wondering how they concluded that

[tex]
N_{h\epsilon}(f(x);d_2) \subseteq N_{\epsilon}(x;d_1)[/tex]
[tex]
N_{\frac{\epsilon}{k}}(f(x);d_1) \subseteq N_{\epsilon}(x;d_2)
[/tex]

Couldn't it also be that

[tex]
N_{h\epsilon}(f(x);d_2) \supseteq N_{\epsilon}(x;d_1)[/tex]
[tex]
N_{\frac{\epsilon}{k}}(f(x);d_1) \supseteq N_{\epsilon}(x;d_2)
[/tex]


Thanks!

micromass

You have proven that if [itex]y\in N_{h\varepsilon}(f(x);d_2)[/itex], then [itex]y\in N_\varepsilon(x;d_1)[/itex]. This implies that [itex]N_{h\varepsilon}(f(x);d_2)\subseteq N_\varepsilon(x;d_1)[/itex].

Indeed, saying that [itex]A\subseteq B[/itex] means exactly that all [itex]y\in A[/itex] also have [itex]y\in B[/itex].

PhysicsGente

Thanks ;)